Hey Chris!

There are many options to consider here.

The most basic way to add some different tones to your setup would be to start adding effects pedals in front of the amp. For example, you could add an overdrive or distortion pedal, set your amp for a clean sound, then kick the pedal on or off to switch between a clean and dirty sound.

The next thing to consider is a multi fx processor. Usually you would want one with foot controls so that you simply stomp on different presets or turn individual effects on or off. There are some relatively inexpensive options here, and you could program all sorts of different tones and add modulation effects like chorus or flanger -- add a delay effect and even reverb.
Units like these allow you to create presets (combinations of effects) that you can switch between, perfect for more radical changes in tone throughout a song.

In more pro setups, guitarists use multi-channel amplifiers (they can switch between clean, crunch, and lead tones) integrated with pedals and multi-fx units that can be controlled by a MIDI foot controller. Some guitarists even switch between different amps (using amp switching devices) to get a variety of tones in a live setting.

Hope this info helps! Let me know if you have more questions.
Mike